29.36.19 problem 1087
Internal
problem
ID
[5646]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
36
Problem
number
:
1087
Date
solved
:
Monday, January 27, 2025 at 01:01:16 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.135 (sec). Leaf size: 272
dsolve(diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^2 = 0,y(x), singsol=all)
\begin{align*}
\int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{{1}/{4}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{{3}/{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} +\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{{1}/{4}}d \textit {\_a} \right )}{\left (y \left (x \right )-a \right )^{{3}/{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{{1}/{4}}d \textit {\_a} \right )}{\left (y \left (x \right )-a \right )^{{3}/{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} +\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{{1}/{4}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{{3}/{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.339 (sec). Leaf size: 369
DSolve[(D[y[x],x])^4 +f[x] (y[x]-a)^3 (y[x]-b)^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x-\sqrt [4]{f(K[1])}dK[1]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x-i \sqrt [4]{f(K[2])}dK[2]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^xi \sqrt [4]{f(K[3])}dK[3]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x\sqrt [4]{f(K[4])}dK[4]+c_1\right ] \\
y(x)\to a \\
y(x)\to b \\
\end{align*}